Directional distributions can have the same normalized mean direction while differing substantially in shape. Comparing them requires a distance whose response to geometric displacement and probability overlap matches the statistical question. We examine ten bounded metrics or metric families for probability measures on the sphere, scaling each to have maximum value one, covering transport, sliced transport, kernel and energy distances, and comparisons based on probability mass. We explain which metrics describe weak convergence, which admit Hilbert-space norm representations, and which pairs attain the maximum distance. Examples and counterexamples show how these distinctions affect empirical comparisons and numerical approximations. A summary table and relationship diagram support the choice of a distance. The discussion separates exact metrics from finite representations, unbiased estimators, and regularized transport costs.

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